Simulation of topological field theories by quantum computers
Abstract
Quantum computers will work by evolving a high tensor power of a small (e.g. two) dimensional Hilbert space by local gates, which can be implemented by applying a local Hamiltonian H for a time t. In contrast to this quantum engineering, the most abstract reaches of theoretical physics has spawned topological models having a finite dimensional internal state space with no natural tensor product structure and in which the evolution of the state is discrete, H = 0. These are called topological quantum filed theories (TQFTs). These exotic physical systems are proved to be efficiently simulated on a quantum computer. The conclusion is two-fold: 1. TQFTs cannot be used to define a model of computation stronger than the usual quantum model BQP. 2. TQFTs provide a radically different way of looking at quantum computation. The rich mathematical structure of TQFTs might suggest a new quantum algorithm.
Cite
@article{arxiv.quant-ph/0001071,
title = {Simulation of topological field theories by quantum computers},
author = {Michael H. Freedman and Alexei Kitaev and Zhenghan Wang},
journal= {arXiv preprint arXiv:quant-ph/0001071},
year = {2009}
}